Abstract

In this paper the expression for the radiative transfer equation (RTE) commonly used when describing light propagation in biological tissues is derived directly from the equation of energy conservation of Maxwell’s equations (Poynting’s theorem) by making use of a volume-averaged expression for the time-averaged flow of energy. The derivation is presented step by step with Maxwell’s equations as the starting point, analyzing all approximations taken in order to arrive at the expression of the scalar RTE employed in biomedical applications, which neglects particle nonsphericity and orientation, depolarization, and coherence effects.

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    [CrossRef]
  4. K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).
  5. E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, 1997).
    [CrossRef]
  6. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol.  1 (Academic, 1978).
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    [CrossRef] [PubMed]
  9. A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23–42 (2006).
    [CrossRef]
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  17. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  18. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (Pergamon, 2006).
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
    [CrossRef]
  20. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
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    [CrossRef]
  23. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499(1994).
    [CrossRef]
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2010 (1)

2009 (1)

2008 (1)

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef] [PubMed]

2006 (1)

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23–42 (2006).
[CrossRef]

2002 (1)

2000 (1)

1998 (1)

N. G. Chen and J. Bai, “Monte carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

1995 (1)

1994 (2)

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Aronson, R.

Arridge, S. R.

Bai, J.

N. G. Chen and J. Bai, “Monte carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chen, N. G.

N. G. Chen and J. Bai, “Monte carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

Dehghani, H.

Ding, K.-H.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Draine, B. T.

Feng, T. C.

Flatau, P. J.

González-Rodríguez, P.

Haskell, R. C.

Hoveiner, J. W.

M. I. Mishchenko, J. W. Hoveiner, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol.  1 (Academic, 1978).

Jacques, S. L.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef] [PubMed]

Kim, A. D.

P. González-Rodríguez, A. D. Kim, “Comparison of light scattering models for diffuse optical tomography,” Opt. Express 17, 8756–8774 (2009).
[CrossRef] [PubMed]

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23–42 (2006).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Mcadams, M. S.

Mishchenko, M. I.

Moscoso, M.

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23–42 (2006).
[CrossRef]

Nieto-Vesperinas, M.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, 1991).

Planck, M.

M. Planck, The Theory of Heat Radiation (P. Blakiston’s Son, 1914).

Pogue, B. W.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef] [PubMed]

Poynting, H.

H. Poynting, Collected Scientific Papers by Henry Poynting (Cambridge University, 1910).

Ripoll, J.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Svaasand, L. O.

Travis, L. D.

M. I. Mishchenko, J. W. Hoveiner, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

Tromberg, B. J.

Tsang, L.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Tsay, T. T.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Yanovitskij, E. G.

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, 1997).
[CrossRef]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

Appl. Opt. (1)

Astrophys. J. (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Inverse Probl. (1)

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23–42 (2006).
[CrossRef]

J. Biomed. Opt. (1)

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (4)

Opt. Express (2)

Phys. Rev. Lett. (1)

N. G. Chen and J. Bai, “Monte carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

Other (13)

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, 1991).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (Pergamon, 2006).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

H. Poynting, Collected Scientific Papers by Henry Poynting (Cambridge University, 1910).

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

M. Planck, The Theory of Heat Radiation (P. Blakiston’s Son, 1914).

M. I. Mishchenko, J. W. Hoveiner, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, 1997).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol.  1 (Academic, 1978).

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Figures (5)

Fig. 1
Fig. 1

Representation of the Poynting vector S ( r ) averaged over the differential volume δ V . This will yield a volume-averaged flow of energy S ( r ) v pointing in an average direction of energy flow s ^ v . The flow of energy at any point in space r has been represented in such a way that it is understood that the Poynting vector is continuous.

Fig. 2
Fig. 2

Representation of the volume-averaged flow of energy S ( r ) v projected onto a generic direction of energy flow s ^ J (compare with Fig. 1).

Fig. 3
Fig. 3

Representation of the surface Σ that encloses the differential volume δ V with a total of N particles distributed with density ρ.

Fig. 4
Fig. 4

Representation of the outward and inward flux, Σ + and Σ , respectively, from surface Σ that encloses the differential volume δ V .

Fig. 5
Fig. 5

Schematic representation of how the inward flux might be considered equivalent to the outward flux through surface Σ that encloses δ V . This has been represented two-dimensionally for a square volume; however, a similar approach can be used for any geometry of Σ.

Equations (85)

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× H ϵ c E t = 4 π c j ,
× E + μ c H t = 0 ,
W t + j · E + c 4 π ( E × H ) = 0 ,
W = 1 8 π ( ϵ E 2 + μ H 2 ) ( J / cm 3 ) ,
d P abs d V = j · E = ω ϵ ( i ) E 2 ( W / cm 3 ) .
S = c 4 π E × H ( W / cm 2 ) .
W t + d P abs d V + · S = 0 ,
S = 1 2 T T T c 4 π [ E ( r , t ) × H ( r , t ) ] d t c 8 π { E 0 × H 0 * } ,
S = ϵ c 0 8 π | E 0 | 2 s ^ ( W / cm 2 ) .
W = 1 8 π ϵ | E 0 | 2 ( J / cm 3 ) .
W = 1 c 0 | S | = 1 c 0 S · s ^ .
d P abs d V = 1 2 ω ϵ ( i ) | E 0 | 2 ( W / cm 3 ) ,
d P abs d V = 4 π ϵ c 0 ω ϵ ( i ) S · s ^ .
1 c 0 S ( r ) · s ^ t + d P abs d V ( r ) + · S ( r ) = 0.
S J ( r ) = S J ( r ) s ^ J = S ( r ) ( s ^ · s ^ J ) s ^ J ,
1 c 0 S ( r ) · s ^ J t + d P abs d V ( r ) ( s ^ · s ^ J ) + s ^ J · ( S ( r ) · s ^ J ) = 0 ,
· S J ( r ) = s ^ J · [ S ( r ) ( s ^ · s ^ J ) ] = s ^ J · ( S ( r ) · s ^ J ) .
w r ( s ^ J ) = 1 δ V S ( r ) v δ V S ( r r ) ( s ^ · s ^ J ) d 3 r ( sr 1 ) ,
S ( r ) v = 1 δ V δ V S ( r r ) d 3 r ( W / cm 2 ) .
s ^ r v = 1 4 π ( 4 π ) w r ( s ^ J ) s ^ J d Ω ,
S ( r ) v = S ( r ) v s ^ r v .
δ V S ( r r ) s ^ · s ^ J d 3 r = δ V S ( r r ) · s ^ J d 3 r = S ( r ) v w r ( s ^ J ) δ V .
2 E 0 ( r ) + k 2 E 0 ( r ) = k 2 [ n 2 ( r ) n 0 ] E 0 ( r ) + [ · E 0 ( r ) ] ,
2 E 0 ( r ) + k 2 E 0 ( r ) = F ( r ) E 0 ( r ) .
E ( sc ) ( r ) = 1 4 π V F ( r ) E 0 ( r ) g ( r , r ) d 3 r ,
E ( sc ) ( r ) f ( s ^ , s ^ 0 ) exp ( i k r ) r ,
f ( s ^ , s ^ 0 ) = 1 4 π V F ( r ) E 0 ( r ) exp ( i k s ^ · r ) d 3 r ,
S ( sc ) = ϵ c 0 8 π | f ( s ^ , s ^ 0 ) | 2 r 2 s ^ .
S ( inc ) V = 1 V V | S ( inc ) ( r ) | d 3 r ,
p ( s ^ , s ^ 0 ) = ϵ c 0 8 π | f ( s ^ , s ^ 0 ) | 2 S ( inc ) V ,
S ( sc ) = S ( inc ) V p ( s ^ , s ^ 0 ) r 2 s ^ .
P ¯ abs = V d P abs d V d V = 1 2 ω V ϵ ( i ) ( r ) | E ( r ) | 2 d 3 r ( W ) .
σ a = P ¯ abs S ( inc ) V = ω 2 S ( inc ) V V ϵ ( i ) ( r ) | E ( r ) | 2 d 3 r ( cm 2 ) .
P ¯ sc = V · S ( sc ) d V ( W ) ,
σ sc = P ¯ sc S ( inc ) V = 1 S ( inc ) V S S ( sc ) · n d S ( cm 2 ) ,
σ sc = S p ( s ^ , s ^ 0 ) d S r 2 .
σ sc = ( 4 π ) p ( s ^ , s ^ 0 ) d Ω ( cm 2 ) ,
p ^ ( s ^ , s ^ 0 ) = p ( s ^ , s ^ 0 ) σ tot ,
E ( sc ) ( r ) = i = 1 N E i ( sc ) ( r ) = i = 1 N f i ( s ^ i , s ^ 0 ) exp ( i k | r r i | ) | r r i | ,
f i ( s ^ i , s ^ 0 ) = 1 4 π V F i ( r i ) E ( r i ) exp ( i k s ^ i · r i ) d 3 r i ,
| E | 2 = ( E ( inc ) ( r ) + i = 1 N E ( sc ) ( r i ) ) ( E ( inc ) ( r ) + i = 1 N E ( sc ) ( r i ) ) * .
S = S ( inc ) + i = 1 N S ( sc ) i + i , j = 1 i j N S ( sc ) i j + ,
S S ( inc ) + i = 1 N S ( sc ) i ,
S ( sc ) ( r ) i = σ tot ( i ) S ( inc ) ( r i ) V p ^ ( s ^ i , s ^ ) | r r i | 2 s ^ i ,
σ ¯ sc = i = 1 N σ sc ( i ) ,
σ ¯ a = i = 1 N σ a ( i ) .
1 c 0 t δ V ( s ^ · s ^ J ) S ( r r ) d 3 r + δ V ( s ^ · s ^ J ) d P abs d V ( r r ) d 3 r + δ V ( s ^ · s ^ J ) s ^ J · r S ( r r ) d 3 r = 0 ,
S S ( inc ) + S ( sc ) .
1 c 0 t δ V S · s ^ J d 3 r 1 c 0 t S ( inc ) v w r ( s ^ J ) δ V ,
δ V S ( inc ) · s ^ J d 3 r δ V S ( sc ) · s ^ J d 3 r ,
δ V d P abs d V ( s ^ · s ^ J ) d 3 r = w r ( s ^ J ) δ V d P abs d V d 3 r N σ a S ( inc ) v w r ( s ^ J ) ,
S ( inc ) ( r ) · s ^ S ( inc ) ( r ) v ,
δ V ( s ^ · s ^ J ) s ^ J · r S ( r r ) d 3 r = w r ( s ^ J ) δ V s ^ J · r S ( r r ) d 3 r .
δ V ( s ^ · s ^ J ) s ^ J · r S ( r r ) d 3 r = w r ( s ^ J ) S ( inc ) · s ^ J + w r ( s ^ J ) δ V s ^ J · r S ( sc ) ( r r ) d 3 r ,
S ( inc ) · s ^ J = 1 δ V δ V s ^ J · r S ( inc ) ( r r ) d 3 r .
S ( inc ) · s ^ J s ^ J · S ( inc ) ,
δ V ( s ^ · s ^ J ) s ^ J · r S ( r r ) d 3 r = w r ( s ^ J ) s ^ J · S ( inc ) ( r ) v δ V + w r ( s ^ J ) δ V s ^ J · r S ( sc ) ( r r ) d 3 r .
δ V s ^ J · S ( sc ) ( r r ) d 3 r = Σ + ( r ) Σ ( r ) ,
Σ + ( r ) = Σ S out ( sc ) ( r r ) s ^ J · s ^ d S ,
Σ ( r ) = Σ S in ( sc ) ( r r ) s ^ J · s ^ d S .
σ sc = 1 S ( inc ) ( r ) v Σ S out ( sc ) ( r r ) s ^ J · s ^ d S ,
δ V s ^ J · S ( sc ) ( r r ) d 3 r = N σ sc w r ( s ^ J ) S ( inc ) ( r ) v Σ ( r ) .
Σ S in ( sc ) ( r r ) s ^ J · s ^ d S = V t s ^ J · S in ( sc ) ( r r ) d 3 r ,     r δ V ,
Σ S in ( sc ) ( r r ) s ^ J · s ^ d S i = 1 N Σ [ S ( sc ) ( r ) i · s ^ i ] s ^ J · s ^ d S     r Σ ,
Σ S in ( sc ) ( r r ) s ^ J · s ^ d S i = 1 N σ tot ( i ) S ( inc ) ( r ) v Σ w r ( s ^ i ) p ^ ( s ^ i , s ^ J ) | r r i | 2 d S .
Σ S in ( sc ) ( r r ) s ^ J · s ^ d S N σ tot S ( inc ) ( r ) v ( 4 π ) w r ( s ^ ) p ^ ( s ^ , s ^ J ) d Ω .
1 c 0 t S ( inc ) v w r ( s ^ J ) + μ a S ( inc ) v w r ( s ^ J ) + s ^ J · [ S ( inc ) v w r ( s ^ J ) ] + μ s S ( inc ) v w r ( s ^ J ) μ t ( 4 π ) S ( inc ) v w r ( s ^ ) p ^ ( s ^ J , s ^ ) d Ω = 0 ,
I ( r , s ^ J ) = 1 4 π S ( r ) v w r ( s ^ J ) ( W cm 2 sr ) ,
I ( r , s ^ J ) = 1 4 π δ V S ( r r ) · s ^ J d 3 r ,
U ( r ) = ( 4 π ) I ( r , s ^ ) d Ω ( W / cm 2 ) ,
U ( r ) = 1 4 π S ( r ) v ( 4 π ) w r ( s ^ ) d Ω = S ( r ) v ,
u ( r ) = 1 c 0 δ V | S ( r r ) | d 3 r = S ( r ) v c 0 ,
u ( r ) = 1 c 0 ( 4 π ) I ( r , s ^ ) d Ω ( J / cm 3 ) .
J ( r ) = ( 4 π ) I ( r , s ^ ) s ^ d Ω ( W / cm 2 ) .
J ( r ) = 1 4 π S ( r ) v ( 4 π ) w r ( s ^ ) s ^ d Ω = S ( r ) v s ^ r v ,
S ( inc ) ( r ) · s ^ S ( inc ) ( r ) v = 1 δ V δ V S ( r r ) d 3 r .
S ( inc ) · s ^ J s ^ J · S ( inc ) v ,
δ V r S ( r r ) d 3 r r δ V S ( r r ) d 3 r ,
1 δ V δ V S ( inc ) · s ^ d V 1 δ V δ V S ( sc ) · s ^ d V .
S ( sc ) i = 1 N S ( sc ) i ,
S S ( inc ) + S ( sc ) ,
μ a = 1 δ V i = 1 N σ a i 1 V t i = 1 N v σ a i ,
μ s = 1 δ V i = 1 N σ sc i 1 V t i = 1 N v σ sc i ,
S ( sc ) i S ( inc ) ( r i ) p ( s ^ i , s ^ ) | r r i | 2 d S .
( · E 0 ) 0.

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